Method and system for determination of molecular interaction parameters

ABSTRACT

A method of determining kinetic parameters for a reversible molecular interaction between a ligand immobilized to a solid support surface and a binding partner to the ligand in solution, comprises sequentially, without intermediate regeneration or renewal of the immobilized ligand, flowing a plurality of fluid volumes containing different known concentrations of the binding partner over the solid support surface, monitoring the momentary amount of binding partner bound to the solid support surface related to time and solution concentration of binding partner and collecting the binding data, and determining the kinetic parameters by globally fitting a predetermined kinetic model for the interaction between the binding partner and the immobilized ligand to the collected binding data, which model allows for mass transport limitation at the solid support surface. An analytical system for carrying out the method, a computer program, a computer program product and a computer system for performing the method are also disclosed.

CROSS REFERENCES TO RELATED APPLICATIONS

This application is a continuation of U.S. application Ser. No.13/659,960 filed Oct. 25, 2012, which is a continuation of U.S.application Ser. No. 13/038,432, now U.S. Pat. No. 8,321,152, which is acontinuation of U.S. application Ser. No. 12/105,355, now U.S. Pat. No.7,925,448, which is a continuation of U.S. application Ser. No.10/861,098, now U.S. Pat. No. 7,373,255, which claims the benefit ofU.S. Provisional Patent Application No. 60/477,909, filed Jun. 12, 2003and U.S. Provisional Patent Application No. 60/526,364, filed Dec. 1,2003; and also claims priority to Swedish Patent Application No.0301639-1, filed Jun. 6, 2003 and Swedish Patent Application No.0303214-1, filed Dec. 1, 2003; all of these applications areincorporated herein by reference in their entireties.

BACKGROUND OF THE INVENTION

Field of the Invention

The present invention relates to the determination of kinetic parametersfor molecular interactions, and more particularly to a method fordetermining kinetic parameters for the interaction between a moleculeimmobilized to a solid support surface and a binding partner to themolecule in solution. The invention also relates to an analyticalsystem, a computer program product and a computer system for performingthe method.

Description of the Related Art

Analytical sensor systems that can monitor interactions betweenmolecules, such as biomolecules, in real time are gaining increasinginterest. These sensor systems, usually referred to as interactionanalysis sensor systems or biospecific interaction analysis sensorsystems, are often based on optical biosensors and affinity analysis andoffer a rapid way to determine in real time inter alia equilibrium andrate constants without the need to label the interacting molecules. Theyhave been used in the study of a variety of biomolecules, includingproteins, nucleic acids, lipids and carbohydrates. In these systems, asensor surface having one of the molecular reactants immobilized theretois contacted with a solution containing the other reactant, either byproviding a flow of the solution past the sensor surface, or in acuvette or the like, and binding interactions at the surface aredetected.

Conventionally, to determine, for example, association and dissociationrate constants (k_(a) and k_(d), respectively) for the interactionbetween two interacting molecules, one of the molecules, often referredto as the ligand, is immobilized to a sensor surface and the othermolecule, often referred to as the analyte, is provided in solution atseveral different known concentrations. Each concentration, or sample,of the analyte is then contacted with the sensor surface, either in alaminar flow past the sensor surface, or in a cuvette or the like, topermit association of the analyte to the sensor surface. After a samplehas been brought to contact the sensor surface, the surface is contactedwith a solution free from analyte, usually buffer, to permitdissociation of the analyte from the immobilized ligand. During theseassociation and dissociation phases, the amount of binding of analyte tothe surface is continuously detected and the binding data is collected.Before contacting the sensor surface with sample of a new analyteconcentration, the ligand surface is restored or “regenerated” bytreating the surface with a regeneration solution capable of removingany bound analyte while not destroying the ligand. In that way, all thedifferent samples will contact essentially one and the same ligandsurface as far as ligand density is concerned. The association anddissociation rate constants can then be obtained from the collectedbinding data by fitting the data to mathematical descriptions ofinteraction models in the form of differential equations. Usually, thebinding data for all the samples are used in the same fit, a procedurereferred to as global fitting. From the determined association anddissociation rate constants k_(a) and k_(d), the equilibrium constant,K_(D), and the affinity constant K_(A) (K_(A)=1/K_(D)) of theinteraction can in turn be calculated. Alternatively, provided that theinteraction reaches steady state during the association phase, theequilibrium constant can be directly obtained from the binding datawithout fitting.

Problems may arise, however, when the ligands are covalently immobilizedto the sensor surface and suitable regeneration conditions are difficultto find. Renewed binding of the ligand via an immobilized capture agentbefore the contacting with each new analyte concentration could then bean alternative, but has the disadvantage of consuming large amounts ofligand for the determination.

Determination of equilibrium constants by a titration procedure withoutthe requirement for regeneration of the immobilized ligands is describedby Schuck, P., et al. (1998) Anal. Biochem. 265, 79-91. The sample iscontinuously circulated in a closed loop over two sensor spots of acommercial surface plasmon resonance biosensor. One of the sensor spotsis functionalized with an immobilized ligand for a soluble analyte inthe sample, and the other sensor spot serves as a reference surface. Abinding isotherm for the interacting molecules is obtained by stepwiseequilibrium titration of the analyte into the circulating loop, i.e. thesensor spots are sequentially contacted with stepwise increasedconcentrations of analyte until equilibrium is attained for eachconcentration. This equilibrium titration is said to be especiallyuseful for the determination of binding constants in high-affinitysystems since it eliminates the need for interpretation of bindingkinetics and thereby problems that may arise from mass transportlimitations.

A similar stepwise equilibrium titration procedure is described for acuvette-based biosensor design in Hall, D. R., and Winzor, D. J. (1997)Anal. Biochem. 244, 152-160.

Also Myszka, D. G., et al. (1998) Anal. Biochem. 265, 326-330 disclosesequilibrium analysis of high affinity interactions using a surfaceplasmon resonance-based biosensor. In this approach, the time availableto collect association phase data is increased by placing the analytedirectly into the running buffer. Complete equilibrium binding profileswere generated without a regeneration step by changing the concentrationof analyte and allowing the surface reactions to reequilibrate. Analyteconcentrations were also decreased to demonstrate that the bindingreactions were fully reversible. In this way, equilibrium dissociationconstants for very high affinity interactions could be determined.

Shank-Retzlaff, M. L., and Sligar, S. G. (2000) Anal. Chem. 72,4212-4220 describes a one-step method for determining kinetic rates andequilibrium binding affinities by a technique called analyte-gradientsurface plasmon resonance SPR (AG-SPR) which eliminates the need forsurface regeneration. A gradient of analyte is passed over the sensorsurface under continuous-flow conditions so that the concentration ofanalyte increases linearly with time. The rate at which analyte binds toimmobilized ligands on the sensor surface is monitored by monitoring thechange in the surface plasmon resonance as the analyte concentrationincreases. Kinetic rates are determined by fitting the data to atwo-compartment model for the molecular interaction which permits usealso for systems under mass transport limitations.

Titration procedures to determine binding capacity and regenerationconditions have been proposed by Karlsson, R., et al., Biasymposium1998, Edinburgh 2-4 Sep. 1998, 5th Biasymposium in Japan, Tokyo, Nov.5-6, 1998.

US-A1-2003/014365 discloses a method for determining interactionparameters, including rate constants, between an analyte and a ligandimmobilized to a sensor surface, where a measurement can be performedseveral times in succession, e.g. in a cuvette, with a stepwisemodification of the analyte concentration each time. The measurementsneed not be completed to equilibrium but can be interrupted earlier andthe concentration of the analyte can be raised or lowered. Separatefitting of the binding curve part corresponding to each analyteconcentration is used to determine respective initial binding rates,from which the association and dissociation rate constants for theinteraction are then determined.

Thus, while titration procedures in combination with biosensors andaffinity analysis are known per se in the art for determiningequilibrium binding affinities, the use of multiple titrations todetermine kinetic rate constants seems to be disclosed only in theabove-mentioned publications Shank-Retzlaff, M. L., and Sligar, S. G.(2000), and US-A1-2003/014365. The method according to Shank-Retzlaff,M. L., and Sligar, S. G. (2000) suffers from the limitation that itrequires the use of a continuous gradient of the analyte. The method ofUS-A1-2003/014365, on the other hand, is disadvantageous in that initialbinding rates are frequently lower than the kinetic binding rate due totransport limitations, making initial binding rates unreliable forkinetic analysis. Further, this type of evaluation restricts theinjection order of analyte concentrations.

From the prior art it may therefore be concluded that for determiningkinetic rates for molecular interactions using systems based onbiosensors and affinity analysis, it is necessary to regenerate theimmobilized ligand prior to contacting the sensor surface with adifferent concentration of analyte to thereby present essentially oneand the same ligand surface to each analyte concentration, unless (i) acontinuous gradient of the analyte is used, or (ii) initial bindingrates are determined in systems free from mass transport limitations.

It is an object of the present invention to provide a sensor-basedmethod for determining chemical interaction parameters, includingkinetic rate constants, by stepwise titration, which method obviatesregeneration procedures while permitting measurements under masstransport limitation.

BRIEF SUMMARY OF THE INVENTION

According to the present invention, it has surprisingly been found thatthe above and other objects and advantages can be achieved by anaffinity sensor-based method for determining chemical interactionparameters, which method comprises performing, in one and the sameexperimental cycle, stepwise changes of the analyte concentrationwithout intermediate regeneration of the sensor surface, and determiningthe interaction parameters using global analysis of the whole set ofdetected binding data with regard to a kinetic model for the molecularinteraction that allows for mass transport limitation, such as e.g. theafore-mentioned two-compartment model. In this method, kinetic data canthus be obtained in spite of mass transport limitations and thedifferent analyte concentrations may be used in any order.

Therefore, in one aspect, the present invention provides a method ofdetermining kinetic parameters for a reversible molecular interactionbetween a ligand immobilized to a sensor surface and a binding partner(analyte) to the ligand in solution, comprising the steps of:

a) sequentially, without intermediate regeneration or renewal of theimmobilized ligand, flowing a plurality of fluid volumes containingdifferent known concentrations of the binding partner over the solidsupport surface to permit association of binding partner to theimmobilized ligand,

b) flowing over the solid support surface a fluid volume free frombinding partner to permit dissociation of binding partner from theligand,

c) monitoring during steps a) and b) the momentary amount of bindingpartner bound to the solid support surface related to time and solutionconcentration of binding partner and collecting the binding data, and

d) determining the kinetic parameters by fitting, preferably globally, apredetermined kinetic model for the interaction between the bindingpartner and the immobilized ligand to the collected binding data, whichmodel allows for mass transport limitation at the solid support surface.

Preferably, the flowing of a plurality of different known concentrationsof binding partner in step a) starts with the concentration of zerobinding partner and continues with concentrations above zero.

In a variant method, the different concentrations of binding partner areobtained by generating a gradient of the binding partner andintersecting segments of the gradient with segments of a fluid free frombinding partner.

In another aspect, the present invention therefore provides a method ofdetermining kinetic parameters for a reversible molecular interactionbetween a ligand immobilized to a sensor surface and a binding partner(analyte) to the ligand in solution, comprising the steps of:

a) sequentially, without intermediate regeneration or renewal of theimmobilized ligand, flowing a plurality of fluid volumes containingdifferent concentrations of the binding partner over the solid supportsurface to permit association of binding partner to the immobilizedligand, wherein the fluid volumes containing binding partner arediscrete segments of a concentration gradient of the binding partnerseparated by segments of fluid free from binding partner,

b) monitoring during step a) the momentary amount of binding partnerbound to the solid support surface related to time and solutionconcentration of binding partner and collecting the binding data, and

c) determining the kinetic parameters by fitting a predetermined kineticmodel for the interaction between the binding partner and theimmobilized ligand to the collected binding data.

Preferably, the model allows for mass transport limitation at the solidsupport surface.

In the case that the concentrations of all the gradient segments are notknown, the unknown concentrations may be estimated. This may, forexample, be done by having the fitting in step d) include local fittingof concentrations.

The solid support surface is preferably a sensor surface, i.e. one whichproduces a detectable signal in response to a binding interaction at thesurface.

The term “monitoring” as used herein means that detection is performedat at least a plurality of times during steps a) and b), preferably at alarge number of times.

In another aspect, the present invention provides a method fordetermining the concentrations of binding partner in a plurality ofbinding partner-containing samples.

In still another aspect, the present invention provides a method fordetermining kinetic parameters for a plurality of different bindingpartners.

In another aspect, the present invention provides an analytical systemfor studying molecular interactions, which system comprises dataprocessing means for performing at least one of the methods.

In yet another aspect, the present invention provides a computer programcomprising program code means for performing at least one of themethods.

In still another aspect, the present invention provides a computerprogram product comprising program code means stored on a computerreadable medium or carried on an electrical or optical signal forperforming at least one of the methods.

In another aspect, the present invention provides a computer systemcontaining a computer program comprising program code means forperforming at least one of the methods.

Other advantages, novel features and objects of the invention willbecome apparent from the following detailed description of the inventionwhen considered in conjunction with the accompanying drawings.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a schematic side view of a biosensor system based on surfaceplasmon resonance (SPR).

FIG. 2 is a representative sensorgram with a binding curve havingassociation and dissociation phases.

FIG. 3 is a representative sensorgram obtained by sequential injectionsof a kinase inhibitor (staurosporin) over a surface with immobilizedkinase according to the method of the present invention.

FIG. 4 is the sensorgram of FIG. 3 having overlaid thereon the bindingcurve resulting from global fitting.

FIG. 5 is a schematic illustration of a system for creating a pulsedconcentration gradient.

FIG. 6 is a schematic illustration of an alternative system for creatinga pulsed concentration gradient.

FIG. 7 is an overlay plot of observed and fitted binding data forsequential injections of a carbonic anhydrase inhibitor (acetazolemide)over a surface with immobilized carbonic anhydrase.

FIG. 8 is a similar overlay plot to that of FIG. 7 for sequentialinjections of another carbonic anhydrase inhibitor (azosulfamide).

FIG. 9 is a similar overlay plot to those of FIGS. 7 and 8 forsequential injections of still another carbonic anhydrase inhibitor(benzenesulfonamide).

FIG. 10 is an overlay plot of observed and fitted binding data forsequential injections of a thrombin inhibitor (melagatran) over asurface with immobilized thrombin.

FIG. 11 is an overlay plot of four binding curves and fitted data, threecurves (to the left) representing sensorgrams with single injections ofanalyte, and one curve (to the right) representing a sensorgram forsequential injections of analyte.

FIG. 12 is a sensorgram obtained by pulse injection of camel antibody(SGS) over a surface with immobilized lysozyme.

DETAILED DESCRIPTION OF THE INVENTION

Unless defined otherwise, all technical and scientific terms used hereinhave the same meaning as commonly understood by a person skilled in theart related to this invention. Also, the singular forms “a”, “an”, and“the” are meant to include plural reference unless it is statedotherwise.

As mentioned above, the present invention relates to the determinationof molecular interaction parameters, including kinetic rate constants,for the interaction between a molecule immobilized to a solid supportsurface and a binding partner to the molecule in solution by a noveltitration type method, preferably in combination with sensor basedtechnology to study the molecular interactions and present the resultsin real time, as the interactions progress. Before describing thepresent invention in more detail, however, the general context in whichthe invention is used will be described.

Chemical sensors or biosensors are typically based on label-freetechniques, detecting a change in a property of a sensor surface, suchas e.g. mass, refractive index, or thickness for the immobilized layer,but there are also sensors relying on some kind of labelling. Typicalsensor detection techniques include, but are not limited to, massdetection methods, such as optical, thermo-optical and piezoelectric oracoustic wave, (including e.g. surface acoustic wave (SAW) and quartzcrystal microbalance (QCM)) methods, and electrochemical methods, suchas potentiometric, conductometric, amperometric andcapacitance/impedance methods. With regard to optical detection methods,representative methods include those that detect mass surfaceconcentration, such as reflection-optical methods, including bothexternal and internal reflection methods, angle, wavelength,polarization, or phase resolved, for example evanescent waveellipsometry and evanescent wave spectroscopy (EWS, or InternalReflection Spectroscopy), both may include evanescent field enhancementvia surface plasmon resonance (SPR), Brewster angle refractometry,critical angle refractometry, frustrated total reflection (FTR),scattered total internal reflection (STIR)—which may include scatterenhancing labels, optical wave guide sensors; external reflectionimaging, evanescent wave-based imaging such as critical angle resolvedimaging, Brewster angle resolved imaging, SPR-angle resolved imaging,and the like. Further, photometric and imaging/microscopy methods, “perse” or combined with reflection methods, based on for example surfaceenhanced Raman spectroscopy (SERS), surface enhanced resonance Ramanspectroscopy (SERRS), evanescent wave fluorescence (TIRF) andphosphorescence may be mentioned, as well as waveguide interferometers,waveguide leaking mode spectroscopy, reflective interferencespectroscopy (RIfS), transmission interferometry, holographicspectroscopy, and atomic force microscopy (AFR).

Commercially available biosensors include the BIACORE® systeminstruments, manufactured and marketed by Biacore AB, Uppsala, Sweden,which are based on surface plasmon resonance (SPR) and permit monitoringof surface binding interactions in real time between a bound ligand andan analyte of interest.

While in the detailed description and Examples that follow, the presentinvention is illustrated in the context of SPR spectroscopy, and moreparticularly the BIACORE® system, it is to be understood that thepresent invention is not limited to this detection method. Rather, anyaffinity-based detection method where an analyte binds to a ligandimmobilized on a sensing surface may be employed, provided that a changeat the sensing surface can be measured which is quantitativelyindicative of binding of the analyte to the immobilized ligand thereon.

The phenomenon of SPR is well known, suffice it to say that SPR ariseswhen light is reflected under certain conditions at the interfacebetween two media of different refractive indices, and the interface iscoated by a metal film, typically silver or gold. In the BIACORE®instruments, the media are the sample and the glass of a sensor chipwhich is contacted with the sample by a microfluidic flow system. Themetal film is a thin layer of gold on the chip surface. SPR causes areduction in the intensity of the reflected light at a specific angle ofreflection. This angle of minimum reflected light intensity varies withthe refractive index close to the surface on the side opposite from thereflected light, in the BIACORE® system the sample side.

A schematic illustration of the BIACORE® system is shown in FIG. 1.Sensor chip 1 has a gold film 2 supporting capturing molecules 3, e.g.antibodies, exposed to a sample flow with analytes 4 (e.g. an antigen)through a flow channel 5. Monochromatic p-polarised light 6 from a lightsource 7 (LED) is coupled by a prism 8 to the glass/metal interface 9where the light is totally reflected. The intensity of the reflectedlight beam 10 is detected by an optical detection unit (photodetectorarray) 11.

A detailed discussion of the technical aspects of the BIACORE®instruments and the phenomenon of SPR may be found in U.S. Pat. No.5,313,264. More detailed information on matrix coatings for biosensorsensing surfaces is given in, for example, U.S. Pat. Nos. 5,242,828 and5,436,161. In addition, a detailed discussion of the technical aspectsof the biosensor chips used in connection with the BIACORE® instrumentsmay be found in U.S. Pat. No. 5,492,840. The full disclosures of theabove-mentioned U.S. patents are incorporated by reference herein.

When molecules in the sample bind to the capturing molecules on thesensor chip surface, the concentration, and therefore the refractiveindex at the surface changes and an SPR response is detected. Plottingthe response against time during the course of an interaction willprovide a quantitative measure of the progress of the interaction. Sucha plot is usually called a sensorgram. In the BIACORE® system, the SPRresponse values are expressed in resonance units (RU). One RU representsa change of 0.0001° in the angle of minimum reflected light intensity,which for most proteins and other biomolecules correspond to a change inconcentration of about 1 pg/mm² on the sensor surface. As samplecontaining an analyte contacts the sensor surface, the ligand bound tothe sensor surface interacts with the analyte in a step referred to as“association.” This step is indicated on the sensorgram by an increasein RU as the sample is initially brought into contact with the sensorsurface. Conversely, “dissociation” normally occurs when the sample flowis replaced by, for example, a buffer flow. This step is indicated onthe sensorgram by a drop in RU over time as analyte dissociates from thesurface-bound ligand.

A representative sensorgram (binding curve) for a reversible interactionat the sensor chip surface is presented in FIG. 2, the sensing surfacehaving an immobilized capturing molecule, or ligand, for example anantibody, interacting with a binding partner therefor, or analyte, in asample. (The detection curves, or sensorgrams, produced by biosensorsystems based on other detection principles mentioned above will have asimilar appearance.) The y-axis indicates the response (here inresonance units, RU) and the x-axis indicates the time (here inseconds). Initially, buffer is passed over the sensing surface givingthe baseline response A in the sensorgram. During sample injection, anincrease in signal is observed due to binding of the analyte. This partB of the binding curve is usually referred to as the “associationphase”. Eventually, a steady state condition is reached where theresonance signal plateaus at C (this state may, however, not always beachieved). At the end of sample injection, the sample is replaced with acontinuous flow of buffer and a decrease in signal reflects thedissociation, or release, of analyte from the surface. This part D ofthe binding curve is usually referred to as the “dissociation phase”.The analysis is ended by a regeneration step where a solution capable ofremoving bound analyte from the surface, while (ideally) maintaining theactivity of the ligand, is injected over the sensor surface. This isindicated in part E of the sensorgram. Injection of buffer restores thebaseline A and the surface is now ready for a new analysis.

From the profiles of the association and dissociation phases B and D,respectively, information regarding the binding and dissociationkinetics is obtained, and the height of the resonance signal representssurface concentration (i.e., the response resulting from an interactionis related to the change in mass concentration on the surface). Thiswill now be explained in more detail below.

Assume a reversible reaction between an analyte A and a surface-bound(immobilized) capturing molecule, or ligand, B which is not diffusion ormass transfer limited and obeys pseudo first order kinetics:A+B

AB

This interaction model (usually referred to as the Langmuir model),which assumes that the analyte (A) is both monovalent and homogenous,that the ligand (B) is homogenous, and that all binding events areindependent, is in fact applicable in the vast majority of cases.

The rate of change in surface concentration of analyte A (=rate ofchange in concentration of formed complex AB) during analyte injectionis the sum of the rates of the analyte A going on and off:

$\begin{matrix}{\frac{d\lbrack{AB}\rbrack}{dt} = {{{k_{a}\lbrack A\rbrack}\lbrack B\rbrack} - {k_{d}\lbrack{AB}\rbrack}}} & (1)\end{matrix}$where [A] is the concentration of analyte A, [B] is the concentration ofthe ligand B, [AB] is the concentration of the reaction complex AB,k_(a) is the association rate constant, and k_(d) is the dissociationrate constant.

After a time t, the concentration of unbound ligand B at the surface is[B_(T)]-[AB], where [B_(T)] is the total, or maximum, concentration ofligand B. Insertion into Equation (1) gives:

$\begin{matrix}{\frac{d\lbrack{AB}\rbrack}{dt} = {{{k_{a}\lbrack A\rbrack}\left\{ {\left\lbrack B_{T} \right\rbrack - \lbrack{AB}\rbrack} \right\}} - {k_{d}\lbrack{AB}\rbrack}}} & (2)\end{matrix}$

In terms of detector response units (AB is detected), this can beexpressed as

$\begin{matrix}{\frac{dR}{dt} = {{k_{a}{C\left( {R_{\max} - R} \right)}} - {k_{d}R}}} & (3)\end{matrix}$where R is the response at time t in resonance units (RU), C is theinitial, or bulk, concentration of free analyte (A) in solution, andR_(max) is the response (in RU) obtained if analyte (A) had bound to allligand (B) on the surface. Rearrangement of Equation (3) gives:

$\begin{matrix}{\frac{dR}{dt} = {{k_{a}{CR}_{\max}} - {\left( {{k_{a}C} + k_{d}} \right)R}}} & (4)\end{matrix}$where R is the response in resonance units (RU). In integrated form, theequation is:

$\begin{matrix}{R = {\frac{k_{a}{CR}_{\max}}{{k_{a}C} + k_{d}}\left( {1 - e^{{- {({{k_{a}C} + k_{d}})}}t}} \right)}} & (5)\end{matrix}$

Now, according to equation (4), if dR/dt is plotted against the boundanalyte concentration R, the slope is k_(a)C+k_(d) and the verticalintercept is k_(a)R_(max)C. If the bulk concentration C is known andR_(max) has been determined (e.g. by saturating the surface with a largeexcess of analyte), the association rate constant k_(a) and thedissociation rate constant k_(d) can be calculated. A more convenientmethod is, however, fitting of the integrated function (5), or numericalcalculation and fitting of the differential Equation (4), preferably bymeans of a computer program as will be described below.

The rate of dissociation can be expressed as:

$\begin{matrix}{\frac{dR}{dt} = {{- k_{d}}R}} & (6)\end{matrix}$and in integrated form:R=R ₀ ·e ^(−k) ^(d) ^(t)  (7)where R₀ is the response at the beginning of the dissociation phase(when the buffer wash of the surface starts).

Equation (6) may be linearized:

$\begin{matrix}{{\ln\left\lbrack \frac{R}{R_{0}} \right\rbrack} = {{- k_{d}}t}} & (8)\end{matrix}$and a plot of ln [R/R₀] versus t will produce a straight line with theslope=−k_(d). More conveniently, however, the dissociation rate constantk_(d) is determined by fitting the exponential rate equation (7).

Affinity is expressed by the association constant K_(A)=k_(a)/k_(d), orthe dissociation constant (also referred to as the equilibrium constant)K_(D)=k_(d)/k_(a).

The association constant K_(A) may alternatively be determined fromEquation (3), where dR/dt=0 at equilibrium, giving:k _(d) R _(eq) =k _(a) C(R _(max) −R _(eq))  (9)where R_(eq) is the detector response at equilibrium. Sincek_(a)/k_(d)=K_(A), insertion in Equation (9) and rearrangement gives:

$\begin{matrix}{\frac{R_{eq}}{X} = {{{- K_{A}}R_{eq}} + {K_{A}R_{\max}}}} & (10)\end{matrix}$

If binding reactions are performed at multiple concentrations, the datamay either be fitted or R_(eq)/C may be plotted versus R_(eq) whichgives the slope=−K_(A). Such an equilibrium analysis may be performedwhen the rates of association and dissociation are too rapid to measureaccurately.

To obtain reliable kinetic constants, the above described analysis isusually repeated for a number of different analyte concentrations and,suitably, also at least one other ligand density at the sensor surface.

Software for the analysis of kinetic and other biosensor data iscommercially available. Thus, for example, evaluation of kinetic dataproduced by the BIACORE® instruments is usually performed with thededicated BIAevaluation software (supplied by Biacore AB, Uppsala,Sweden) using numerical integration to calculate the differential rateequations and non-linear regression to fit the kinetic parameters byfinding values for the variables that give the closest fit reducing thesum of squared residuals to a minimum. The residuals are the differencebetween the calculated and the experimental curve at each point, squaredresiduals being used to weight equally deviations above and below theexperimental curve. The sum of squared residual is expressed by Equation(11):

$\begin{matrix}{S = {\sum\limits_{1}^{n}\;\left( {r_{f} - r_{x}} \right)^{2}}} & (11)\end{matrix}$where S is the sum of squared residuals, r_(f) is the fitted value at agiven point, and r_(x) is the experimental value at the same point.

For example, for the molecular interaction described above, suchsoftware-assisted data analysis is performed by, after subtractingbackground noises, making an attempt to fit the above-mentioned simple1:1 Langmuir binding model as expressed by Equations (5) and (7) aboveto the measurement data.

Usually the binding model is fitted simultaneously to multiple bindingcurves obtained with different analyte concentrations C (and/or withdifferent levels of surface derivatization R_(max)). This is referred toas “global fitting”, and based on the sensorgram data such globalfitting establishes whether a single global k_(a) or k_(d) will providea good fit to all the data. The results of the completed fit ispresented to the operator graphically, displaying the fitted curvesoverlaid on the original sensorgram curves. The closeness of the fit isalso presented by the chi-squared (χ²) value, a standard statisticalmeasure. For a good fitting, the chi-squared value is in the samemagnitude as the noise in RU². Optionally, “residual plots” are alsoprovided which give a graphical indication of how the experimental datadeviate from the fitted curve showing the difference between theexperimental and fitted data for each curve. The operator then decidesif the fit is good enough. If not, the sensorgram or sensorgramsexhibiting the poorest fit are excluded and the fitting procedure is runagain with the reduced set of sensorgrams. This procedure is repeateduntil the fit is satisfactory.

Sometimes, the above-mentioned 1:1 binding reaction model will not bevalid, which requires the data set to be reanalysed using one or moreother reaction models. Such alternative models may include, for example,a one to one reaction influenced by mass transfer (mass transport), twoparallel independent one to one reactions, two competing reactions, anda two state reaction. Parallel reactions can occur when the immobilizedligand is heterogeneous, whereas a heterogenous analyte may give rise tocompeting reactions. A two state reaction indicates a conformationchange that gradually leads to a more stable complex between ligand andanalyte. For differential rate equations reflecting these alternativereaction models, it may be referred to, for example, Karlsson, R., andFält, A. (1997) J. Immunol. Methods 200, 121-133 (the disclosure ofwhich is incorporated by reference herein). For a more comprehensivedescription of curve fitting with regard to the BIACORE® system, it maybe referred to the BIAevaluation Software Handbook (Biacore AB, Uppsala,Sweden) (the disclosure of which is incorporated by reference herein).

As regards mass transport, transport effects will influence the kineticsof binding when the reaction rate is fast compared to the rate oftransport. For a reaction where analyte (A) binds to immobilized ligand(B), binding with mass transport limitation may be represented by thereaction formula:A _(solution)

A _(surface) +B

AB

The differential equations describing the binding interaction willtherefore include terms for mass transfer of analyte to the surface. Fora flow cell, a “two-compartment” model consisting of a set of coupledordinary differential equations and described in inter alia Myszka, D.G., et al. (1998) Biophys. J. 75, 583-594, and Shank-Retzlaff, M. L.,and Sligar, S. G. (2000) supra (the relevant disclosures of which areincorporated by reference herein) is considered to give a reasonabledescription of the binding kinetics when the data are influenced by masstransport. In this model, the flow cell is assumed to be divided intotwo compartments, one in which the concentration of analyte is constant,and a second near the sensor surface where the analyte concentrationdepends on the mass transport rate, the surface density of ligand, andthe reaction rate constants.

For the interaction of a monovalent analyte (A) reacting with animmobilized monovalent ligand (B), this model may be represented by thefollowing two differential equations (12) and (13):

$\begin{matrix}{\frac{dA}{dt} = \left( {{{- k_{a}}{A\left( {B_{T} - {AB}} \right)}} + {k_{d}{AB}} + {k_{i}\left( {C - A} \right)}} \right)} & (12) \\{\frac{dAB}{dt} = {{k_{a}{A\left( {B_{T} - {AB}} \right)}} - {k_{d}{AB}}}} & (13)\end{matrix}$where k_(t) is the transport coefficient describing diffusive movementof analyte between the compartments, B_(T) is the total ligandconcentration, A is the bulk concentration of free analyte, C is theinjection (i.e. initial) analyte concentration, AB is the concentrationof complex AB (=surface density of bound analyte), and k_(a) and k_(d)are the association and dissociation rate constants, respectively.

It is understood that the use of this two-compartment model analyzingthe kinetic globally will extend the range of reaction rates that can bedetermined with instruments like the BIACORE® and has also beenincorporated in the above-mentioned BIAevaluation analysis software(available from Biacore AB, Uppsala, Sweden).

Theoretical interaction data for different interaction models may besimulated using a simulator, such as e.g. the simulator which isbuilt-in in the above-mentioned BIAevaluation analysis software. In thisway, the effects of varying experimental parameters may be explored,thereby helping to determine which interaction conditions that may becrucial in distinguishing between alternative kinetic models.

The Invention

As mentioned above, determining kinetic parameters using, for example, aBIACORE® instrument, conventionally comprises monitoring the interactionof analyte with immobilized ligand for a plurality of different analyteconcentrations with regeneration, or renewal, of the immobilized ligandbefore each measurement with a new analyte concentration. A kineticmodel for the interaction is then fitted globally to the collectedbinding data, usually in the form of sensorgrams, to give the kineticparameters.

According to the present invention, such a flow cell-based analysis fordetermining kinetic parameters may be substantially improved and speededup by a titration type, or “sequential injection”, procedure wherein theligand-supporting surface is successively contacted, in one and the sameanalytical cycle, with the different analyte concentrations to produce acontinuous sensorgram. A kinetic interaction model allowing for masstransport limitation, such as the two-compartment model mentioned above,is then fitted globally to the entire sensorgram to calculate thekinetic parameters. In addition to being considerably lesstime-consuming, this titration type method reduces the amount ofexperimental data to be evaluated and eliminates the risk ofregeneration conditions destroying immobilized ligand.

The order of introducing the different analyte concentrations is notcritical to the success of the inventive procedure. Rather, the analyteconcentrations may, in fact, be introduced in any order. For example,the analyte concentrations may be successively increased, orsuccessively decreased, or may alternately be higher and lower, etc.Further, the same concentration of analyte may be introduced repeatedly,if desired.

The injections of analyte, which preferably may be short, for a BIACORE®or similar instrument e.g. in the order of 30-60 seconds, may beinterrupted when an increased analyte concentration only leads to amarginally increased response. This may advantageously be effected usingso-called adaptive software. The formed complex is then preferablyallowed to dissociate for a longer period of time, e.g. from about 5 to60 minutes in the context of a BIACORE® or similar instrument, to addadequate dissociation data for the kinetic evaluation. Such adissociation period may, optionally, instead (or additionally) beperformed between any two analyte injections in the cycle.

It is to be noted that the design of the above-mentioned BIACORE®instruments, for example, requires that each injection of analyte befollowed by injection of running buffer, permitting partial dissociationof the bound analyte, before the next concentration of analyte isinjected. It is understood that such alternating analyte-free liquidvolumes, for example with about one minute intervals, will adddissociation data to the those obtained by the above-mentioned(preferably longer) dissociation period. Provided that sufficientdissociation data is obtained by these alternating injections ofanalyte-free fluid, the longer dissociation period may even be dispensedwith. Intermediate buffer injections may, of course, also be used, evenif a sensor instrument design as such permits sequential injections ofanalyte without buffer injections between them.

At least for some applications, it is preferred that the flow issubstantially constant during the entire experimental cycle.

The sequential injection procedure may optionally be repeated at leastonce with the same or another sequence of analyte concentrations, andall the collected binding data may then be fitted simultaneously to thekinetic interaction model.

Optionally, the fluid volumes containing the different analyteconcentrations may be different volume segments of a concentrationgradient of the analyte. A method for the sequential injection ofsegments of a decreasing (or increasing) analyte gradient is describedin our copending U.S. application entitled “Method and apparatus forcharacterization of interactions” (the disclosure of which isincorporated by reference herein).

In this “pulse injection” method, each injection consists of a series ofshort sample pulses, suitably 4 or 5 up to 40 pulses, preferably 15-30,more preferably about 20 pulses, generated by alternating flows ofanalyte-containing sample, and another liquid, such as buffer. Eachpulse preferably has a volume of 1-40 μl, preferably 10-40 μl, morepreferably 15-25 μl, suitably about 20 μl. The duration of the pulses,i.e. each segment of solution can be 8-20, preferably about 10-15,suitably 12 seconds long, and the flow rate for the sample liquidthrough the flow cell may be 50-200, preferably 80-120, suitably 100μl/min.

In this way, information from several concentration levels will begenerated through a single injection in that each pulse of the injectionin principle constitutes one concentration. Optionally, some pulsesduring an injection may be discarded, whereby the discarded segmentswill not be passed through the sensor. Alternatively, some aliquot(s) ofliquid can be discarded even before performing the alternating bufferinjections to create the separated segments.

Depending on how the concentration gradient is prepared, theconcentrations of all segments may or may not be known. If allconcentrations are known, the kinetic parameters may be determined byglobal fitting as described above. If, on the other hand, only theconcentration of, say, an edge segment(s) is known, such as in the caseof a “dispersion gradient”, the fitting may have to include localfitting of the analyte concentrations.

If desired, binding data from one or more sequential experimentsperformed as described above may be combined with binding data from oneor more conventional type single concentration experiments, and thecombined data are then fitted to the kinetic interaction model. Thus,generally, the total data set that is globally fitted to the kineticinteraction model may comprise any combination of binding curvesincluding one or more “sequential” binding curves and one or more singleconcentration binding curves.

To facilitate the understanding of the invention, reference is now madeto FIG. 3 which shows an exemplary sensorgram (as obtained with theabove-mentioned BIACORE® instrument) for a titration with successivelyincreased concentrations of an analyte (A) reacting with an immobilizedligand (B) to form a complex (AB) which gives rise to the detected SPRresponse. This titration is described in more detail in Example 1 below.Referring to the sensorgram, analyte concentration 80 nM was injectedbetween 404-464 seconds, 240 nM between 523-583 seconds, 730 nM between642-702 seconds, and 2200 nM between 760-820 seconds.

Before analysing the binding curve, start and end times for eachinjection of analyte are identified, and the base line is adjusted tozero level. Generally, the binding curve may then be analysed using aninteraction model based on time dependent analyte concentration andallowing for mass transport limitation, such as the above-mentionedtwo-compartment model. If necessary, this model may be adapted to otherinteraction mechanisms, such as e.g. conformation change (A+B

AB

AB*), competing reactions (A1+B

A1B; A2+B

A2B), parallel reactions (A+B1

AB1; A+B2

AB2), or bivalent analyte (A+B

AB; AB+B

AB2). Such models are described in more detail in, for example, theBIAevaluation Software Handbook mentioned above.

Assuming, for example, 1:1 binding, the following differential equationmay represent the time dependent concentration of analyte A capable ofbinding to immobilized ligand B.

$\begin{matrix}{\frac{dA}{dt} = {{k_{t}\left( {C_{1} + C_{2} + C_{3} + {\ldots C}_{n} - A} \right)} - \left( {{k_{a}A*B} - {k_{d}{AB}}} \right)}} & (14)\end{matrix}$where A is the surface concentration of free analyte A, k_(t) is atransport coefficient, C₁ to C_(n) are the different bulk concentrationsof analyte, B is the concentration of immobilized ligand (liganddensity), AB is the concentration of formed complex (=bound analyte),and k_(a) and k_(d) are the association rate constant and dissociationrate constant, respectively.

If at the beginning of the experiment the analyte concentration, A, atthe surface is assumed to be zero, the concentration of ligand, B, isassumed to be R_(max), and the concentration of formed complex, AB, isassumed to be zero, the following functions, here written in a formcompatible with the above-mentioned BIAevaluation software, describe theinteraction system.AB+RI1*$1+RI2*$2+RI3*$3+RI4*$4+RI5*$5;$1=(sign(t−tOn1)−(sign(t−tOff1))/2;$2=(sign(t−tOn2)−(sign(t−tOff2))/2;$3=(sign(t−tOn3)−(sign(t−tOff3))/2;$4=(sign(t−tOn4)−(sign(t−tOff4))/2;$5=kt*($1*conc1+$2*conc2+$3*conc3+$4*conc4−A);$6=ka*A*B−kd*AB;A=$5-$6|0;B=−$6|Rmax;AB=$6|0

In the above model, sign functions $1 to $4 (being 1 or 0) indicate whena certain analyte concentration is injected (tOn1 is the time whenanalyte concentration 1 is injected, tOff1 is the time when analyteconcentration 1 is stopped, etc). Functions $5 and $6 describe togetherthe variation of the surface concentration of analyte over time, $6describing how the ligand density and complex varies with time. Thefunction on the first line corresponds to the total change in signalfrom binding AB and bulk effects present during each injection. Bynumerical integration of these equations and global fitting to all datapoints of the whole sensorgram in FIG. 3, using e.g. BIAevaluationsoftware (Biacore AB, Uppsala, Sweden) adapted to the present invention,the kinetic rate constants k_(a) and k_(d) can be obtained. Mathematicalmodels for the other interaction mechanisms may readily be devised by aperson skilled in the art.

The result of the fit is shown in FIG. 4, where the solid “non-ripplingline” indicates the adapted curve for best fit for the concentrationrange 80-2200 nM, k_(a) and k_(d) being constrained to single values of1.0*10⁵ M⁻¹s⁻¹ and 6.6*10⁻⁴ s⁻¹, respectively, as described in Example 1below.

With the above model, kinetic and affinity data can thus be obtained inspite of transport limitations. Since, as mentioned above, the method isinsensitive to the injection order of analyte, analysis of relativelylow analyte concentrations may be performed when binding is close tosaturation or above the equilibrium response, thereby maximizing thekinetic information in the sensorgram where transport effects are small.

Data from binding analyses at two or more different ligand densities mayadvantageously be fitted simultaneously.

Measurement of concentration simultaneously with kinetic rate constantsmay be performed, using the above titration procedure and data fitting,provided that the transport coefficient k_(t) is known and keptconstant. Preferably, binding data from at least two, and preferablymore ligand densities are then used.

A similar titration type approach as those described above may also beused to determine (at least approximate) kinetic rate constants for aplurality of different analytes binding to the same ligand (at the sameor different known concentrations). In this case, different analytes aresequentially passed over the sensor surface rather than differentanalyte concentrations. The evaluation of the binding data is modifiedto permit fitting of different rate constants to the resulting totalbinding curve. Such a procedure may, for example, be used to screendifferent binding partners towards the ligand to quickly find the bestbinder, e.g. drug candidates binding to a drug target, such as areceptor. Optionally, this approach may be combined with the use ofdifferent analyte concentrations.

As mentioned above, analyte injections may also be performed by a “pulseinjection” approach. An exemplary system for performing such a pulseinjection method, based on the microfluidic system of a BIACORE® 3000instrument (Biacore AB, Uppsala, Sweden), is schematically illustratedin FIG. 5.

As can be seen in FIG. 5, there are provided two vessels (e.g. testtubes) 20 and 21 containing sample (20) and buffer (21), respectively.There is also provided a means 22 for aspirating liquid from the testtubes, indicated with vertical lines extending down into the test tubes.This means 22 can suitably be a needle, and since the same needle isused for both liquids, the needle shown in the sample tube 20 is shownby a broken line. The needle would thereby be physically moved betweenthe tubes for the aspiration of liquids sequentially. Of course, thereare other possibilities of devising the aspiration means, the one shownbeing only exemplary.

A system buffer supply is also provided. Initially the entire system isfilled with buffer, i.e. all tubing contains this buffer. Respectivesegments of tubing 23 (sample and system buffer, respectively) arecoupled to a microfluidic device 24 (in the BIACORE® instrumentsreferred to as an Integrated Fluidic Cartridge—IFC) enabling controlledliquid delivery to one or more flow cells, in the illustrated case fourflow cells 25 a to 25 d. Each flow cell has a sensor surface onto whichone or more suitable target(s) are immobilized. There are also provideda number of valves v1 to v4 in the microfluidic device 24 for thecontrol of the flows of the respective liquids. Valves v2 and v3 controlthe supply of fluid to the flow cells whereas valves v1 and v4 each areconnected to waste. The valves v1 to v4 are controlled by a control unit26. Alternatively, the flow in the various lines can be controlled byaccurate pumps, whereby the actual flow rates can be monotonicallycontrolled to provide the desired flow rates, ranging from zero flow tothe maximum flow rates required, or combinations thereof.

The first step in the procedure is to aspirate a small volume of bufferinto the needle 22, i.e. to immerse the needle into the buffer tube 21,and to aspirate the appropriate volume into the needle. It is howevernot strictly necessary to fill the needle with buffer by aspiration.Instead, the needle can be filled with buffer from the other end, i.e.from the system buffer supply by filling the entire system with buffer.Then, the needle is moved to the sample tube and a suitable volume ofsample of about 500 μl is aspirated. However, the actual volume maydepend on the application and the kind of sample, and can vary withinwide limits, say between 1 μl and 4 ml.

The aspiration of sample will lead to mixing of the sample and buffer bydispersion, thereby creating a gradient in the tubing. In this case, thegradient will be a decreasing gradient (as seen from the needle) runningthrough the tubing. If an increasing gradient is required, one wouldhave to aspirate buffer after the sample aspiration, and ensure that anon-dispersed sample trailing edge is provided by first aspirating anair bubble to protect the sample from liquid already present in theneedle, second a sample and third a buffer segment. The aspirationsequence always ends with aspiration of one or a few air bubbles toprotect the gradient from liquid already present in the microfluidicdevice 24.

Prior to the first step, it is preferable to perform a few alternatingair and sample aspirations to provide consecutive segments of air andsample and to inject them into the microfluidic device 24. In this waythe sample will be protected from unwanted dispersion with runningbuffer in the microfluidic device, i.e. the leading front of theaspirated sample liquid will exhibit the nominal (maximum)concentration.

When a gradient has been established, it is injected via the needle intothe microfluidic device and valves v2 and v3 in the sample and bufferflow lines are opened and closed according to a programmed sequence toenable alternating sample (exhibiting a gradient in the longitudinaldirection of the tubing) and buffer pulses to be fed into the flowcells, such that the sample liquid flow is intersected at least once,preferably a plurality of times, by a further liquid, represented by thesystem buffer in this case. This intersection will create at least twoseparated segments of liquid. However, other further liquids than thesystem buffer are of course possible, such as pure solvent, solutionscontaining other species of interest, etc.

Thus, the leading edge of a decreasing sample gradient flow willrepresent a first concentration. Most often the concentration at theleading edge will be very close to the nominal, and can be taken torepresent a known concentration. However, the major part of the sampleflow will exhibit a gradient, and thus the majority of said segmentsthat are created will have different concentrations with respect to thesample.

After a predetermined volume of sample gradient flow has passed into theflow cells, valve v2 is closed and valve v3 is opened, thereby injectingbuffer into the line behind the sample flow. During the passage ofanalyte-containing sample over the sensor surface having targetsimmobilized on it, the analyte will associate with the targets. Thevolume of sample should preferably be sufficient to enable equilibriumto establish. However, it is not always required that equilibrium bereached but an equilibrium level can be calculated from thecorresponding binding curve graph (response vs time). The time framesinvolved depend on sample specific binding and transportcharacteristics, flow rate, temperature, flow cell dimensions, etc.

When sample has been injected for a sufficiently long time, buffer isinjected by opening valve v3 and closing valve v2. During the passage ofbuffer over the surface, analyte will dissociate. The process isrepeated until the aspirated sample has been injected.

It is not necessary to inject the complete gradient into the flow cell.During buffer injection (v3 open, v2 closed) valve 1 can be opened todiscard a small segment of the gradient. This will reduce the number ofpulses produced and reduce the time needed for a full injection.

In an alternative embodiment, wherein a system without valves is used,during the association phase, i.e. during the time that the sample ispassed through the sensor cell, the buffer flow is set to a very lowvalue, less than 5%, and e.g. about 1% of the regular flow. This is notstrictly necessary, but prevents sample solution from leaking into thebuffer line. Then a certain, predetermined amount of sample is injectedinto the microfluidic device at a specified rate. The buffer flow rateis then reset to the regular rate. During the passage of buffer throughthe cell, sample compound that has bound to the target on the sensorsurface is allowed to dissociate for a suitable time period.

In FIG. 6 there is shown an alternative method of creating a gradientusing the microfludic device in FIG. 5. This is done by aspirating asample segment of known concentration and diluting it with buffer in themicrofluidic device using a connection c1 and a tubing segment ml. Thisallows the buffer and sample to form a homogenous mixture prior tocontacting the flowcells. The pulses would be generated as previouslydescribed, i.e. by using alternating pumps or valves v2 and v3. Theconnection c1 could be a simple T-connection so that the concentrationof the sample in the gradient is controlled by how the ratio of[flowrate (buffer)] and [flowrate (sample)] changes over time. Anotherpossibility could be to have a two-way valve as connection c1. Theconcentration of the sample will be controlled by switching the inlet tom1 between buffer and sample, having the two-way valve open for buffer adifferent time than open for sample. In the tubing segment m1 thediscrete connected segments of sample and buffer will form a homogenousmix due to dispersion. This method makes it possible to generate agradient with known concentration of the sample at all times, incontrast to the dispersion concentration gradient where only the firstfew pulses have a known compound concentration. In the latter case, itwill be necessary to fit the concentrations locally simultaneously asfitting the kinetic parameters. This will be described in more detailbelow.

While it currently is preferred to use so-called label-free detectionmethods, including those mentioned above, such as SPR, other detectiontechniques may, as already mentioned above, of course, also becontemplated, such as those based on the detection of a label, e.g. aradiolabel, fluorophore (including surface fluorescence), chromophore,chemiluminescer, a marker for scattering light, an electrochemicallyactive marker, a thermoactive marker, etc.

The methods of the invention, including titration (i.e. sequentialanalyte injections) and evaluation with time dependent analyteconcentration, are, as already mentioned above, suitably reduced topractice in the form of a computer system running software whichimplements the different steps of the procedure. Preferably, thetitration is adaptive as mentioned above. The invention also extends tocomputer programs, particularly computer programs on or in a carrier,adapted for putting the titration procedure of the invention intopractice. The carrier may be any entity or device capable of carryingthe program. For example, the carrier may comprise a storage medium,such as a ROM, a CD ROM or a semiconductor ROM, or a magnetic recordingmedium, for example a floppy disc or a hard disk. The carrier may alsobe a transmissible carrier, such as an electrical or optical signalwhich may be conveyed via electrical or optical cable or by radio orother means. Alternatively, the carrier may be an integrated circuit inwhich the program is embedded.

Software like that outlined above may also be used to generatetheoretical interaction curves for different interaction modelssimilarly as described further above for the conventional singleinjection analysis procedures. By generating data for a complex model,for instance a conformation change model, varying parameters such asinjection time and analyte concentration and then fitting this data toboth a 1:1 binding model and a conformational change model, theresiduals can identify conditions for optimal designs of kineticexperiments.

In the following Examples, various aspects of the present invention aredisclosed more specifically for purposes of illustration and notlimitation.

EXAMPLES

Instrumentation

A BIACORE® S51 instrument (Biacore AB, Uppsala, Sweden) was used inExamples 1 to 4 below. This instrument has two Y-type flow cells whichallow a dual flow of fluids over a sensor chip surface, so-calledhydrodynamic addressing, as described in, for example, EP-B1-1021703(the disclosure of which is incorporated by reference herein). Theinstrument uses three parallel detection spots on the sensor chip.

A BIACORE® 3000 instrument (Biacore AB, Uppsala, Sweden) was used inExample 5 below. In this instrument, a micro-fluidic system passessamples and running buffer through four individually detected flow cells(one by one or in series).

As sensor chip was used Series S Sensor Chip CM5 (Biacore AB, Uppsala,Sweden) which has a gold-coated surface with a covalently linkedcarboxymethyl-modified dextran polymer hydrogel.

The outputs from the instruments via the instrument control software are“sensorgrams” which are a plots of detector response (measured in“resonance units”, RU) as a function of time. An increase of 1000 RUcorresponds to an increase of mass on the sensor surface ofapproximately 1 ng/mm². In Examples 1 to 4, evaluation was performedusing BIAevaluation software, version 3.1 (Biacore AB, Uppsala, Sweden),adapted to a kinetic model as described above. In Example 5, evaluationwas performed using BIAevaluation software, version 3.1 (Biacore AB,Uppsala, Sweden), Matlab version 5.3 (The MathWorks, Inc. Natick, Mass.,U.S.A.) and Excel 97 (Microsoft Corp., Redmond, Wash., U.S.A.).

Example 1 Binding of a Kinase Inhibitor to Immobilized Kinase

Anti-histidine antibody (Qiagen 34660; Qiagen, Venlo, Netherlands) wasdiluted to 20 μg/ml in 10 mM acetate buffer pH 5.0, and 13000 RU ofantibody was immobilized to a Series S Sensor Chip CM5 using aminecoupling (Amine Coupling Kit—Biacore AB, Uppsala, Sweden) according tothe manufacturer's instructions. The flow was 10 μl/min. Histidinetagged kinase (proprietary) was then captured and cross-linked to thesensor chip surface by injecting 30 μg/ml of the kinase in 10 mM PBS forthree minutes followed by 2.5 min injections of EDC/NHS and ethanolamineto stabilise the kinase on the surface. With this procedure 4000 RU ofkinase were immobilized.

A kinase inhibitor, staurosporin, was injected at concentrations of 80,240, 730 and 2200 nM at a flow of 30 μl/min. The assay buffer was 50 mMTris pH 7.5, 150 mM NaCl, 10 mM MgCl₂ and 3% DMSO. Overlay plots ofobserved and fitted data for the injection sequence are demonstrated inFIG. 3 (observed) and FIG. 4 (observed plus fitted). The associationrate constant was calculated to 1.0*10⁵ M⁻¹ s⁻¹, and the dissociationrate constant k_(d) to 6.6*10⁻⁴ s⁻¹.

Example 2 Binding of a Set of Carbonic Anhydrase Inhibitors toImmobilized Carbonic Anhydrase

Carbonic anhydrase (Sigma C3640), 20 μg/ml in 10 mM acetate buffer pH5.0, was immobilized to a Series S Sensor Chip CM5 using amine couplingaccording to the manufacturer's instructions (flow 10 μl/min), to obtainsensor surfaces with approximately 2000 and 700 RU, respectively, ofimmobilized carbonic anhydrase. Kinetic analysis was performed byinjecting inhibitors in 10 mM PBS buffer with 0.005% P20 and 5% DMSO ata flow of 30 μl/min. The inhibitors were acetazoleamide (Sigma A6011),azosulfamide (Sigma A2759), and benzenesulfonamide (Aldrich 10,814-6).Overlay plots demonstrating observed and fitted data for the two liganddensities are presented in FIG. 7 (acetazoleamide), FIG. 8(azosulfamide) and FIG. 9 (benzenesulfonamide). In FIG. 7, datarepresent two repeats on each ligand density, and k_(a) was calculatedto 1.8*10⁶ M⁻¹ s⁻¹, and k_(d) to 4.0*10⁻² s⁻¹. In FIG. 8, k_(a) wascalculated to 2.9*10⁴ M⁻¹ s⁻¹, and k_(d) to 8.1*10⁻³ s⁻¹, and in FIG. 9k_(a) was calculated to 1.6*10⁵ M⁻¹ s⁻¹, and k_(d) to 1.4*10⁻¹ s⁻¹.

Example 3 Binding of a High Affinity Inhibitor to Immobilized Thrombin

Thrombin (Sigma T 1063), 20 μg/ml in 10 mM acetate buffer pH 5.0, wasimmobilized to a Series S Sensor Chip CM5 using amine coupling accordingto the manufacturer's instructions (flow 10 μl/min), resulting in sensorsurfaces with approximately 2300 and 600 RU, respectively, ofimmobilized thrombin. Kinetic analysis was performed by injecting a highaffinity thrombin inhibitor, melagatran (a gift from Johanna Deinum,AstraZeneca, Mölndal, Sweden), in 10 mM PBS buffer with 0.005% P20, 5%DMSO and 3.4 mM EDTA at a flow of 30 μl/min. Overlay plots demonstratingobserved and fitted data are presented in FIG. 10. The calculated k_(a)was 2.0*10⁷ M⁻¹ s⁻¹, and k_(d) 1.8*10⁻² s⁻¹.

Examples 1 to 3 above demonstrate that a wide range of rate constantscan be determined with the sequential injection procedure. In theseExamples k_(a) values range from 2.9*10⁴ to 2.0*10⁷ M⁻¹ s⁻¹ and k_(d)values range from 0.14 to 6.6 10⁻⁴ s⁻¹. The Examples also illustratethat kinetic data can be obtained using one or two ligand densities andthat the injection order can be mixed and not only in the order from lowto high concentration.

Example 4 Combined Sequential and Single Injections

In FIG. 11, simulated data representing sequential and single injectionsare combined in the evaluation where all binding curves are fittedsimultaneously. The figure shows an overlay plot of four binding curvesand fitted data. The three curves to the left represent sensorgrams withsingle injections of analyte at 51200 nM but with varying injectiontimes, 10 s, 30 s and 500 s, whereas the curve to the right representssequential injections of analyte from 800 to 12800 nM. This illustratesthat sensorgrams with varying numbers of injections and injection timescan be analysed.

Example 5 Estimation of Interaction Rate Constants

The kinetics of a camel derived heavy chain triple mutant single domainantibody (cAb-Lys3:s SGS) binding to lysozyme (obtained from theDepartment of Ultrastructure, Vrije Universiteit, Brussels, Belgium) wasstudied with the “pulse injection method” described above. Allexperiments were performed at 30° C. 190 RU (chip 1) and 280 RU (chip 2)of lysozyme was immobilized using a standard amine coupling procedure.Upon a 2-min activation, lysozyme (8 μg/ml in 10 mM Na₂HPO₄, pH 7.0) wasinjected for 3 min (chip 1) and 4 min 30 s (chip 2). 10 mM Na₂HPO₄, pH7.0 (flow rate 5 μl/min) was used as running buffer duringimmobilization. SGS was injected at different initial concentrations(0.5, 1.0 and 2.0 μM in HBS-EP). The results are shown in FIG. 12.

Bulk errors in the sample solutions were corrected for by subtraction ofthe reference flow cell signals. Individual pulses were separated andaligned, using MATLAB, so that each pulse corresponded to one bindingcurve. The curves were superimposed in the BIAevaluation software. 15pulses were used in every fit. The first two pulses were assumed to beof initial concentration. Global starting values of k_(a), k_(d) andR_(max) were fitted to the second pulse (pulse number one was omittedbecause of its irregular shape), since its concentration was known.These values were then used to locally fit the concentrations of allpulses. k_(a), k_(d) and R_(max) estimations were refined, using the newconcentration information. The process was repeated until all parametersconverged. Each pulse injection was evaluated separately. The fitting ofthe concentration resulted in a partially linear concentration gradient.Kinetic data obtained with the pulse injection method is presentedtogether with mean values and standard deviations in Table 1 below.

TABLE 1 Results from a pulse injection assay with camel antibody SGS andlysozyme C₀(μM)* k_(a) (M⁻¹ s⁻¹⁾ k_(d) (s⁻¹) R_(max) (RU) K_(D) (M) χ²**Chip 1 0.5 1.75e5 0.508 137  2.9e−6 0.0972 (190 RU) 0.5 9.99e4 0.435 1804.35e−6 0.151 0.5 1.17e5 0.484 170 4.14e−6 0.122 1 1.91e5 0.475 112 2.5e−6 0.243 1  4.6e5 0.514 72.7 1.12e−6 0.198 1 1.62e5 0.513 1263.17e−6 0.325 Chip 2 1 4.60e5 0.466 119 1.01e−6 0.383 (280 RU) 1 1.11e50.462 286 4.15e−6 0.374 1 1.32e5 0.42  251 3.19e−6 0.348 2 1.87e5 0.487188 2.60e−6 1.53 2 2.41e5 0.464 151 1.93e−6 0.816 2 2.79e5 0.421 1371.51e−6 1.85 Average: 2.18e5 0.471 133 189 2.71e−6 St. dev: 1.25e5 0.03339.3 66.8 1.16e−6 Rel. 57% 7% 30% 35% 43% st. dev.: *C₀ is the nominalconcentration of SGS **χ² is a statistical measure of the quality of thefit

It is to be understood that the invention is not limited to theparticular embodiments of the invention described above, but the scopeof the invention will be established by the appended claims.

The invention claimed is:
 1. An analytical system for monitoringinteractions between molecules comprising: an interaction detector fordetecting a reversible molecular interaction between i) a ligandimmobilized to a solid support surface and ii) a binding partnercontacting the ligand in a flow of a fluid at the support surface; amicrofluidic system arranged to provide the flow of the fluid to theinteraction detector support surface, the microfluidic system furthercomprising: a sample solution inlet for introducing said binding partnerinto the system; a buffer solution inlet; a tubing segment fluidicallyconnecting the sample solution inlet and the buffer solution inlet tothe interaction detector, and a controller arranged to control a supplyof a sample solution and a buffer solution into the tubing segment andto feed a gradient of the sample solution in a longitudinal direction ofthe tubing segment and thereby to supply a gradient flow of the fluidcomprising the binding partner to the interaction detector supportsurface.
 2. The analytical system as claimed in claim 1, wherein thecontroller supplies the gradient flow of the fluid as a plurality offluid volumes containing different concentrations of the binding partnerto the interaction detector support surface.
 3. The analytical system asclaimed in claim 2, wherein said different concentrations are applied tothe interaction detector sequentially, without regeneration or renewalof the immobilized ligand.
 4. The analytical system as claimed in claim2, wherein the controller provides for flowing the plurality of fluidvolumes containing different concentrations over the solid supportsurface to permit association of the binding partner to the immobilizedligand, wherein the different concentrations are discrete segments of aconcentration gradient of the binding partner separated by segments offluid free from the binding partner.
 5. The analytical system as claimedin claim 2, wherein the at least two different concentrations arestepped dilution changes applied to the interaction detector.
 6. Ananalytical system employing surface plasmon resonance (SPR) formonitoring real time interactions between molecules comprising: an SPRinteraction detector comprising a solid support surface and an opposingsurface, for detecting a reversible molecular interaction between i) aligand immobilized to the solid support surface and ii) a bindingpartner contacting the ligand in a flow of a fluid at the supportsurface; a microfluidic system arranged to provide the flow of the fluidto the interaction detector support surface, the microfluidic systemcomprising: a sample solution inlet for introducing said binding partnerinto the system; a buffer solution inlet; a tubing segment fluidicallyconnecting the sample solution inlet and the buffer solution inlet tothe interaction detector, and a controller arranged to control a supplyof a sample solution and a buffer solution into the tubing segment andto feed a gradient of the sample solution in a longitudinal direction ofthe tubing segment and thereby to supply a plurality of fluid volumescontaining different concentrations of the binding partner to theinteraction detector support surface.